• Title, Summary, Keyword: Lucas sequence

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SUM FORMULAE OF GENERALIZED FIBONACCI AND LUCAS NUMBERS

  • Cerin, Zvonko;Bitim, Bahar Demirturk;Keskin, Refik
    • Honam Mathematical Journal
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    • v.40 no.1
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    • pp.199-210
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    • 2018
  • In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.

ON THE k-LUCAS NUMBERS VIA DETERMINENT

  • Lee, Gwang-Yeon;Lee, Yuo-Ho
    • Journal of applied mathematics & informatics
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    • v.28 no.5_6
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    • pp.1439-1443
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    • 2010
  • For a positive integer k $\geq$ 2, the k-bonacci sequence {$g^{(k)}_n$} is defined as: $g^{(k)}_1=\cdots=g^{(k)}_{k-2}=0$, $g^{(k)}_{k-1}=g^{(k)}_k=1$ and for n > k $\geq$ 2, $g^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n-2}+{\cdots}+g^{(k)}_{n-k}$. And the k-Lucas sequence {$l^{(k)}_n$} is defined as $l^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n+k-1}$ for $n{\geq}1$. In this paper, we give a representation of nth k-Lucas $l^{(k)}_n$ by using determinant.

New Approach to Pell and Pell-Lucas Sequences

  • Yagmur, Tulay
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.23-34
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    • 2019
  • In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

FIBONACCI AND LUCAS NUMBERS ASSOCIATED WITH BROCARD-RAMANUJAN EQUATION

  • Pongsriiam, Prapanpong
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.511-522
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    • 2017
  • We explicitly solve the diophantine equations of the form $$A_{n_1}A_{n_2}{\cdots}A_{n_k}{\pm}1=B^2_m$$, where $(A_n)_{n{\geq}0}$ and $(B_m)_{m{\geq}0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques [9] and L. Szalay [13] concerning a variant of Brocard-Ramanujan equation.

(±1)-INVARIANT SEQUENCES AND TRUNCATED FIBONACCI SEQUENCES OF THE SECOND KIND

  • CHOI GYOUNG-SIK;HWANG SUK-GEUN;KIM IK-PYO
    • Journal of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.761-771
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    • 2005
  • In this paper we present another characterization of (${\pm}1$)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence $x\;{\in}\;R^{\infty}$ is (-1)-invariant(l-invariant resp.) if and only if $D[_x^0]$ is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where $$D=diag((-1)^0,\; (-1)^1,\;(-1)^2,{\ldots})$$.

ON CONDITIONALLY DEFINED FIBONACCI AND LUCAS SEQUENCES AND PERIODICITY

  • Irby, Skylyn;Spiroff, Sandra
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.1033-1048
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    • 2020
  • We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

A COMPLETE FORMULA FOR THE ORDER OF APPEARANCE OF THE POWERS OF LUCAS NUMBERS

  • Pongsriiam, Prapanpong
    • Communications of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.447-450
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    • 2016
  • Let $F_n$ and $L_n$ be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides $F_k$. Marques obtained the formula of $z(L^k_n)$ in some cases. In this article, we obtain the formula of $z(L^k_n)$ for all $n,k{\geq}1$.